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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-6071be339fb7262c"]},{"id":191278,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-6071be339fb7262c"]},{"id":191279,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"48911973fb102038a69bc96662a913137bb5fedb8c871d3524f18c03eccc614d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2d05bfc763190112","role":"sequence","name":"Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:        0;\n  1:        1;\n  2:       23,           2;\n  3:      206,         412,            3;\n  4:     1677,       12682,         5031,             4;\n  5:    13452,      266902,       400353,         53808,             5;\n  6:   107659,     4807224,     17748693,       9614448,        538295,            6;\n  7:   861322,    80048136,    602387961,     803183948,     200120340,      5167932,           7;\n  8:  6890633,  1276302958,  17625610338,   46750323452,   29376017230,   3828908874,    48234431,         8;\n  9: 55125128, 19845666178, 470925473655, 2207410963968, 2759263704960, 941850947310, 69459831623, 441001024, 9;\n","recurrence_sentence":"The row polynomials satisfy (7*n-6)*P(n,x) = -(343*n+49)*x^2*P(n-4,x) - ((98*n+14)*x+(686*n^2-1862*n+63)*x^2)*P(n-3,x) - ((343*n+49)*x^2-(343*n+49)*x^3)*d/dx P(n-3,x) - (56*n+8+(98*n^2-364*n+44)*x+(343*n^3-1568*n^2+1932*n-720)*x^2)*P(n-2,x) - ((49*n+7)*x+(343*n^2-637*n+245)*x^2-(343*n^2-588*n+252)*x^3)*d/dx P(n-2,x) + (63*n-47+(98*n^2-266*n+156)*x)*P(n-1,x) + ((49*n-42)*x-(49*n-42)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 23 + 2*x, P(3,x) = 206 + 412*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 23 + 2*x;\nP[3, x_] := 206 + 412*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(343*n + 49)*x^2*P[n - 4, x] - ((98*n + 14)*x + (686*n^2 - 1862*n + 63)*x^2)*P[n - 3, x] - ((343*n + 49)*x^2 - (343*n + 49)*x^3)*D[P[n - 3, x], x] - (56*n + 8 + (98*n^2 - 364*n + 44)*x + (343*n^3 - 1568*n^2 + 1932*n - 720)*x^2)*P[n - 2, x] - ((49*n + 7)*x + (343*n^2 - 637*n + 245)*x^2 - (343*n^2 - 588*n + 252)*x^3)*D[P[n - 2, x], x] + (63*n - 47 + (98*n^2 - 266*n + 156)*x)*P[n - 1, x] + ((49*n - 42)*x - (49*n - 42)*x^2)*D[P[n - 1, x], x])/(7*n - 6)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 23 + 2*x; v[4] = 206 + 412*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(343*n + 49)*x^2*v[i - 4] - ((98*n + 14)*x + (686*n^2 - 1862*n + 63)*x^2)*v[i - 3] - ((343*n + 49)*x^2 - (343*n + 49)*x^3)*deriv(v[i - 3], x) - (56*n + 8 + (98*n^2 - 364*n + 44)*x + (343*n^3 - 1568*n^2 + 1932*n - 720)*x^2)*v[i - 2] - ((49*n + 7)*x + (343*n^2 - 637*n + 245)*x^2 - (343*n^2 - 588*n + 252)*x^3)*deriv(v[i - 2], x) + (63*n - 47 + (98*n^2 - 266*n + 156)*x)*v[i - 1] + ((49*n - 42)*x - (49*n - 42)*x^2)*deriv(v[i - 1], x))/(7*n - 6)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2d05bfc763190112","steps":["D"],"edges":["e-6071be339fb7262c"],"text":"A144445 \u2192 D"},"headline":"A144445 \u2192 D","chain":null}